# GLOBAL EXISTENCE FOR 3D NAVIER-STOKES EQUATIONS IN A THIN PERIODIC DOMAIN

• Kwak, Min-Kyu (DEPARTMENT OF MATHEMATICS, CHONNAM NATIONAL UNIVERSITY) ;
• Kim, Nam-Kwon (DEPARTMENT OF MATHEMATICS, CHOSUN UNIVERSITY)
• Received : 2011.05.20
• Accepted : 2011.06.05
• Published : 2011.06.25

#### Abstract

We consider the global existence of strong solutions of the 3D incompressible Navier-Stokes equations in a thin periodic domain. We present a simple proof that a strong solution exists globally in time when the initial velocity in $H^1$ and the forcing function in $L^p$(0,${\infty}$;$L^2$), $2{\leq}p{\leq}{\infty}$ satisfy certain condition. This condition is basically similar to that by Iftimie and Raugel[7], which covers larger and larger initial data and forcing functions as the thickness of the domain ${\epsilon}$ tends to zero.

#### Acknowledgement

Supported by : Chonnam national university

#### References

1. R.A. ADAMS, "Sobolev Spaces", Academic Press, New York, 1975.
2. J.D. AVRIN, Large-eigenvalue global existence and regularity results for the Navier-Stokes equations, J. Differential Equations 127(1996), 365-390. https://doi.org/10.1006/jdeq.1996.0074
3. P. CONSTANTIN AND C. FOIAS, "Navier-Stokes equations", University of Chicago Press, Chicago, 1988.
4. H. FUJITA AND T. KATO, On the Navier-Stokes initial value problem, Arch. Rational Mech. Anal. 16(1964), 269-315. https://doi.org/10.1007/BF00276188
5. E. HOPF, Uber die Anfangswertaufgabe fur die hydrodynamischen Grudgleichungen, Math. Nachr. 4(1951), 213-231.
6. D. IFTIMIE, The 3D Navier-Stokes equations seen as a perturbation of the 2D Navier-Stokes equations, Bull. Soc. Math. France 127(1999), 473-517. https://doi.org/10.24033/bsmf.2358
7. D. IFTIMIE AND G. RAUGEL, Some results on the Navier-Stokes equations in thin 3D domains, J. Differential Equations 169(2001), 281-331. https://doi.org/10.1006/jdeq.2000.3900
8. D. IFTIMIE, G. RAUGEL, AND G.R. SELL, Navier-Stokes equations in thin 3D domains with Navier boundary conditions, Indiana univ. Math. J. 56(2007), 1083-1156. https://doi.org/10.1512/iumj.2007.56.2834
9. M. KWAK AND N. KIM, Remark on global existence for 3D Navier-Stokes equations in Lipschitz domain, Submitted (2007).
10. I. KUKAVICA AND M. ZIANE, Regularity of the Navier-Stokes equation in a thin periodic domain with large data, Discrete and continuous dynamical system 18(2006), 67-86.
11. I. KUKAVICA AND M. ZIANE, On the regularity of the Navier-Stokes equation in a thin periodic domain, J. Differential Equations 234(2007), 485-506. https://doi.org/10.1016/j.jde.2006.11.020
12. J. LERAY, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math. 63(1934), 193-248. https://doi.org/10.1007/BF02547354
13. S. MOMGTGOMERY-SMITH, Global regularity of the Navier-Stokes equations on thin three dimensional domains with periodic boundary conditions, Electronic J. Diff. Eqns. 11(1999), 1-19. https://doi.org/10.1023/A:1021889401235
14. G. RAUGEL AND G.R. SELL, Navier-Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions, J. Amer. Math. Soc. 6(1993), 503-568.
15. G.R. SELL AND Y. YOU, "Dynamics of Evolutionary Equations", Applied Math. Sciences 143, Springer, Berlin, 2002.
16. R. TEMAM, "Navier-Stokes equations and nonlinear functional analysis", CBMS Regional Conference Series, No. 66, SIAM, Philadelphia, 1995.
17. R. TEMAM AND M. ZIANE, Navier-Stokes equations in three-dimensional thin domains with various boundary conditions, Adv. Differential Eqations 1(1996), 499-546.